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Global Existence of Small Solutions for Cubic Quasi–linear Klein–Gordon Systems in One Space Dimension

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Abstract

In this paper, we consider a system of two cubic quasi–linear Klein–Gordon equations with different masses for small, smooth, compactly supported Cauchy data in one space dimension. We show that such a system has global existence when the nonlinearities satisfy a convenient null condition. Our results extend the global existence proved by Sunagawa recently under the non–resonance assumption to that under the resonance assumption.

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References

  1. Klainerman, S.: Global existence of small amplitude solutions to nonlinear Klein–Gordon equations in four space-time dimensions. Comm. Pure Appl. Math., 38, 631–641 (1985)

    MATH  MathSciNet  Google Scholar 

  2. Shatah, J.: Normal forms and quadratic nonlinear Klein–Gordon equations. Comm. Pure Appl. Math., 38, 685–696 (1985)

    MATH  MathSciNet  Google Scholar 

  3. Moriyama, K., Tonegawa, S., Tsutsumi, Y.: Almost global existence of solutions for the quadratic semilinear Klein–Gordon equation in one space dimension. Funkcialaj Ekvacioj, 40, 313–333 (1997)

    MATH  MathSciNet  Google Scholar 

  4. Delort, J. M.: Existence globale et comportement asymptotique pour l’équation de Klein–Gordon quasilinéaire à données petites en dimension 1. Ann. Sci. École Norm. Sup., 34, 1–61 (2001)

    MATH  MathSciNet  Google Scholar 

  5. Moriyama, K.: Normal forms and global existence of solutions to a class of cubic nonlinear Klein–Gordon equations in one space dimension. Diff. Int. Equations, 10, 499–520 (1997)

    MATH  MathSciNet  Google Scholar 

  6. Sunagawa, H.: On global small amplitude solutions to systems of cubic nonlinear Klein–Gordon equations with different mass terms in one space dimension. J. Diff. Eqn., 192, 308–325 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  7. Delort, J. M., Fang, D. Y., Xue, R. Y.: Global existence of small solutions for quadratic quasilinear Klein-Gordon systems in two space dimensions. J. Funct. Anal., 211(2), 288–323 (2004)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Zhejiang University, Hangzhou 310027, P. R. China

    Dao Yuan Fang & Ru Ying Xue

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  1. Dao Yuan Fang
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  2. Ru Ying Xue
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Correspondence to Dao Yuan Fang.

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Supported by NSF of China 10271108 and DFG

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Fang, D.Y., Xue, R.Y. Global Existence of Small Solutions for Cubic Quasi–linear Klein–Gordon Systems in One Space Dimension. Acta Math Sinica 22, 1085–1102 (2006). https://doi.org/10.1007/s10114-005-0668-4

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  • DOI: https://doi.org/10.1007/s10114-005-0668-4

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